Tits has proved that a finite simple group $G$ with a [BN-pair][1] of rank $n \ge 3$, is of Lie type. Let $B$ be the Borel subgroup and $(W,S)$  Coxeter system. The subset lattice of the set $S$ is 1-1 with the interval $[B,G]$ of parabolic subgroups, so is equivalent to the boolean lattice of rank $n$.

We ask about the following extension of Tits' theorem:  
Is a finite simple group $G$ having a subgroup $H$ with $[H,G]$ boolean of rank $n \ge 3$, of Lie type?

For $|G| < 4 \cdot 10^6$ (using GAP) the only examples are ${\rm A}_3(2)$, $^2{\rm A}_2(5^2)$, ${\rm C}_3(2)$ and $^2{\rm A}_3(3^2)$.

  [1]: https://en.wikipedia.org/wiki/(B,_N)_pair